710 / 1042
G
E
o
M
figure; anJ fineem h
tri~n¡;le
is equal tOthe proJu.'\ of
1r.1If the ble into the perpen,lico!.If, it is evideot that the
fum of alt the
tfI.o~l<s tu~ethcr,
th"t is the polygoo, is
cqu.1 to the
produ~t
of
h.lfthe fumof the b,&s (th"t
i,
the half uf Ihe circ\llllfmnce of Ihe polygon) iotu Ihe
COmntoo perpeodic\lLIf heighl of Ihe triaoglcsdrawo from
the ceolre C
lO
ooe of Ihe liJes ; for ex,mple, lO
All.
PR O P O S I T I ON XXX II!.
Fl c. 16.
n,
arta of aei,elcir f o"nd hy mll/lip" .
i7lg
Ih, hal!
of
li" puipbuy inlo Ih, ,adiuJ or Ih,
;'01[
of
Ih, ,adiuJ inloIh, peripbay
-fol a cirele is001d,f–
[ereol fromaoordioale or regular polygon of ao iolioue
number of fides, and ,he commoo heighl of Ihe Iri.ogles
iOlo whieh Ihe polygoo or cirele may be fuppored lO be
dividcd is Ihe radius of Ihe cirele.
Were il wonh while, il were cafy lOdemoollrale
ac–
curalely Ihis propofilioo, by meaos of lhe iofcribe" Qod
circum/Cribed figures, as is dooe io Ihe ¡th prop. of Ihe
u eatire of ¡\rchimedes coocerniog Ihe dimeofioos of lhe
, irele.
CORO LL .4 Rr.
Heoce alroil "l'pears, Ihat Ihe ,reaof Ihefeaor ABCD
ia produced by OIultiplying Ihe half of Ihe
are
inlo Ihe
n dius, and likewife lhat Ihe arca of the fegOleot of lhe
(irele ADC is found by fubtra8ing Irom Ihe
area
of Ihe
fe80r Ihe area of Ihe Iriangle ABC.
PR O P O S [ T [ O N XXXIV.
Flc. 17.
r he eirele
iJ
111h, [qua"
of
Ih, diam'ler,
ar 11
lo
14
morlj.- For
if Ihe diameler AB be fuppo·
f~d
10
be 7, Ihe circumfereoee AHBK u'ill be . Imoll 22
(by Ihe nd prop. of Ihis pan), aod the area of Ihe
fquar~
DC wiIJ be 49; and, by Ihe preeeding. prop, Ihe
arca of Ihe tirde will be
38~ :
therefore Ihe lr¡uare DC
will be 10 Ihe iofcribed cirele as
49
lO 38}, or as 98 lO
77,
Ihal is, as 14 lOI
J.
~
E. D.
If grcater exa/ln, fs is required, you m, y procred 10
any degree of aceuraey: for Ihe fq uare DC is lOIhe io–
fclibed cirele,
as
110 l-t
+
+-
~
+
~
-,'r
+
T'T'
'6e. in infinil.!».
.. This feries will be of 00 ferviee for compuling Ihe
.. area of Ihe eirele aecurately, wilhoUI fome funh er ar–
.. lifiee, becaufe il cenverges at 100 1I0w a rale.
1
he
.. area of the eirele will be fóuod exa81y enough for
.. mofl purpores, by multiplying Ihe fqum of Ihe dia·
" meler by 7354, .nd dil iJing by 10,000, or cUlllng
.. off four deeim.1plaees from Ihe proJu<'\; for Ihe arca
.. of the eirele is In Ihe eircumfcllbed fquare nearly as
.. 7854 lO10.000."
P RO P O S I T ION XXXV.
Flc. 18.
r ofind Ih, are.
of
a given ,I/ipft
- Lel
ABCO bc an e,l hpfe, whoíe grealer diameler is BD, aod
Ihe leffer AC, b,fe,9ing Ihe grealer
perpendi~ularly
in E.
Let a mean proponion.1li F be found (by I31h 6 Eucl. )
belweeo AC aod BO, aod (by Ihe 33d of Ihis) fina
the arca of the eirde dcCc" bed on Ihe diammr HF.
'fhis area is rqn,d lO the arta of Ihe , IIipli: AI;CD.
Fol' becacfe, as I:D
10
AC, fo Ihe fq u.trC uf BD
tu
Iho
fquare 01 HF, (by
2.
eor 20lh
6.
EneJ.): bUI (hy Ihe
cd 12. Euel .) ,s Ihe fq uare of \l D
10
Ihe f'lu,,,, of HF,
,fo is Ihe cirele of Ihe diarneter !J D 10 Ihe eirele of Ihe
,i!iamerer
H
f : Iherefore as
Jj
O
lO
A
C, fo is Ihe ei,
el,
of
f.
T
Y.
Ihe di.lmem
li D
tO Ihe cirele of the di'r.1eler
HF. And
(by lhe 51h plOp.
,l
AlclIlme.dlS of f"he,o,ds)
as
the
g ..
wer J",n,cter
/l D
10 Ihe lene, AC, lo
I!
the cirele of
Ihe di "nel<f
1:0
_10 the ell'pf. ABClJ. Cnnfequently
(by the 11th
5.
Eucl. ) Ihe
wel.
of Ihe diarnel<f
BD
will have Ihe lame proponion lo Ihe eirele of Ihe diame–
ler
J!f,
ano lOIhe , II'pfe ABCD. Therefore, (by
9
th
5,
h d.)
Ihe arca of Ih, ell el. of Ihe diameler
HF
will
be cllu.11O Ihc arca of Ihe eJliple ABCD.
~
E. D.
S
e
H
o
L I U
/1'1.
FromthislodIhe tWopreceding propofilioos,
a
rpelhodis
denl'edof linding Ihe are. ofan eJliple.There aretwo ways:
tll,
Say,
as
one
IS
tuIhe lenerdiamcI<f, foisIhe grealer
U".
meler lO' foun hnum"er, (wh,eh is found by Ihe rule of
thlee.) Then aS"n f. y,
a.
14 tOII ,foislhe4lhoumber
fouodlO
I~e
are. luught. HUllhefeeond wayisfhoner. Mul–
tiply Ihe I. ff. r Jiammr tolOIhe grealer, and Ihe produ/!
by 11; Iheo d" ide Ihe whole produ8 by 14, and Ihe
quolienl wdl be Ihe are. foughl of Ihe ,IIlpre. For ex–
ample, Lel Ihe gre.ler d"mcler be 1
O,
and Ihe lefI'er 7;
br
multiplyiog 10 by 7, Ihe p,odu8 is 70; .od muh,–
p ying Ihal by
1
1, il is 770 ; and uividiog 770 by 14, Ihe
quollem will
be 55,
which is Ihe area of Ihe ellipfe
lought.
" 1
he area of Ihe ellipfe will be fouod more aecurale·
.. Iy,
by multiplyiog Ihe produ8 of Ihe IWOdi.meltr3
.. by 7854'"
We Ih. II add 00 more . boul olher plaio furfaces, whe–
Iher ,,8· IIOear or eurvilinear, whieh feldom oecur iD
pra/!iee ; bUI fh all fu bjoio fome propofilioos .bou! mea–
lu riog the furfaees of folids.
P RO P
O
S [ T
ION
XXXVI.
r o IItrafur, Ih, fuiface
l'
nny pri[III .- Hy
Ihe t41h
defioiliooof Ihe IIlh Enel. a prifm iseoolainedby planes,
of whieh llIiO oppofi le fides (commoolycalled Ihe bafes)
are plain reélilioell figu'es ; ",hich are eilher regular and
ordinal<, and OIeafurcd by prop.
32.
of Ihis; or howevrr
irreg..Ju, and Ihen Ihey arc mcafured by Ihe 331h prop.
The ol her fides are parallelogra Ols, which are Oleafured
by prop. 281h; aod Ihe whole fu perneies of the prrliu
coofifl s of Ihe fum of Ihofe lakeo . ltogelher.
l'
R O P
O
S I T
I ON
XXXVII.
r o
1I,(11u"
Ih, fupufiei"
of
. ny pyralllid.-Sioee
ils
bdfis is • re8ilioe., figure, aoJ the ren of Ihe plaoes te' –
OlinallOg io II'e 10p of Ihe pyramid are triangles; Ihefe
m<afurru f'par.llcly, and .dJed togelher, give Ihe fur–
faee of Ihe pyramid required.
I'
R O P O S I T
I ON Xx.,XVIII.
r o
/f/'Qfrm
Ih,
fr,perfidtl
of
. nJ
"',~uIJf
h'il&.–
Thcfe bodies are " IIe,! regular, which are bounded by
equd.lml and equi.ogu lar figures. The fuperfieies of
Ihé tet raedron eonfills of four equal and equiangular Iri–
angb; Ihe fuperficies of Ihe hexaedron, or cube, of
r.x
e'lu,11fquares ; . n oéledron, of eighl equdl equil' leral
I mn~les ;
a doueeaed<on, of Iwelve equJI .od ordioale
penl,'gons ; "ud Ihe fupcrneies of ao ieofixdroo, of
I<ven'y <qual and equil' le,,' Iriaogles. Th<rdore il
" di be
<aJÍ'
lOTll"fure Ihefe furfarel fromwhal has becn
, Irrady n,o"o.
[n Ihe
(,,"C
manner <ve m"y ole.fure Ihe furerfieirs of
a fo:iJ
eunl.un,o by any
pl,~es.
PR0-